The manipulation of statistical formulas is no substitute
for knowing what one is doing.
 Hubert M Blalock Jr

Chapter 10
    Sec 10.2a

Tests of Significance:

Example 10.9   page 560

Diet colas use artificial sweeteners to avoid sugar.  Colas with artificial sweeteners gradually lose sweetness over time.  Manufacturers therefore test new colas for loss of sweetness before marketing them.  Trained tasters sip the cola along with drinks of standard sweetness and score the cola on a “sweetness score” of 1 to 10.  The cola is then stored for a period of time, then each taster scores the stored cola.  This is a matched pairs experiment.  The reported data is the difference in tasters’ scores.  The bigger the difference, the bigger the loss in sweetness.

 

            2.0       0.4       0.7       2.0       -0.4     2.2       -1.3      1.2       1.1       2.3

The sample mean  indicates a small loss of sweetness.

Consider that a different sample of tasters would have resulted in different scores, and that some variation in scores is expected due to chance.

Does the data provide good evidence that the cola lost sweetness in storage?

To answer that question, we will perform a significance test.

1.   Identify the parameter.

      The parameter of interest is m, the mean loss in sweetness.

2.   State the null hypothesis.

      There is no effect or change in the population.  This is the statement we are trying to find evidence against.

      The cola does not lose sweetness.

     

      State the alternative hypothesis.

      There is an effect or change in the population.

      This is the statement we are trying to find evidence for.

The cola does lose sweetness.

     

3.   Calculate a statistic to estimate the parameter.

      Is the value of the statistic far from the value of the parameter?  If so, reject the null hypothesis.  If not, accept the null hypothesis.

4.   Calculate the P-value. 

If it is small, your result is statistically significant.

Suppose the individual tasters’ scores vary according to a normal distribution with mean  and .  We want to test the null hypothesis so we assume .

 

So the sampling model for  is approximately normal with mean  and standard deviation

N(0, 0.316)

 

 

 

                 -0.9      -0.6    -0.3      0     0.3      0.6     0.9

 

Our sample mean, , was 1.02.  Assuming that the null hypothesis is true, what is the probability of getting a result at least that large? 

Normalcdf ( 1.02, 1E99, 0, 0.316 ) = 0.0006

The probability to the right of  is called the P-value.

The P-value is 0.0006, meaning that we would only expect to get this result in 6 out of 10,000 samples.  This is very unlikely, so we will reject the null hypothesis in favor of the alternative hypothesis and conclude that the cola actually did lose sweetness.

If the P-value is small we say that our result is statistically significant.  The smaller the P-value, the stronger the evidence provided by the data.

How small is small enough?  Compare the P-value to the value of the significance level a.  This value is usually predetermined.

If the P-value is as small or smaller than a, we say that the data are statistically significant at level a.

Hypotheses can be one-sided or two-sided.

        is a one-sided hypothesis because we are only looking at one direction, greater than.

        is a two-sided hypothesis because we are looking at two directions, greater than and less than.

Index